It seems only the VS 2013 Remote tools for ARM works on Windows 8.1.
Does this mean that Visual Studio 2012 is obsolete now for WinRT development?
Yes, it's indeed obsolete.
For windows 8.1 RT development, set up a separate 8.1 machine with VS 2013.
As of Visual Studio 2012 Update 4, remote debugging to 8.1 seems to be available.